Summary of Proposal, 2007

In the space provided below, state the objectives of the proposed
research program and summarize the scientific approach, highlighting
the novelty and expected significance of the work to a field or fields
in the natural sciences and engineering. [...] Your summary must not
exceed forty-five lines on the printed copy.

Over the next five years I plan
to pursue following three projects.

* Algebraic Knot Theory. For many years
now the Kontsevich integral Z (a universal finite type
invariant of knots and links) is appreciated for its strength. It is
stronger than all known "quantum invariants" taken together. But only
recently I understood that that might not be where the real power of
this invariant lies: Z beautifully extends to an invariant of
knotted trivalent graphs which is well behaved under certain natural
operations defined on graphs - edge deletion, "unzipping" and connected
sums. Several well known and hard-to-detect properties of knots are
"definable" using these operations, including the knot genus,
unknotting numbers and the property of being a ribbon knot. Thus, at
least in principle, each of these properties can be translated to a
simple algebraic "equation" involving Z of the knot being
studied. But turning this principle into results is a five-year
project. We still have to identify and study appropriate quotients of
the target space of Z which make Z computable to all
orders. Every classical knot polynomial (Alexander's, Jones', etc.)
defines such a quotient, and there are many other quotients to choose
from, but even the simplest of these quotients, corresponding to a
canonical extension of the Alexander polynomial to graphs, is poorly
understood. See http://www.math.toronto.edu/~drorbn/Talks/Aarhus-0706.
* Knot Theoretic Algebra. My paper "On
Associators and the Grothendieck-Teichmuller Group" indicates
strongly that the right context for understanding Drinfel'd's theory of
formal associators is a certain category of braid groups and operations
mapping such braids groups to each other. There is strong evidence that
the theory of quantum groups (specifically, quantum universal
enveloping algebra and/or quasi-triangular Hopf algebras) should be
related in a similar manner to virtual knots and braids and operations
among them. Indeed, one of the starting points of the theory of quantum
groups is the quantization of Lie bialgebras, and the "universal"
diagrammatic theory underlying Lie bialgebras is the same as the
diagrammatic theory that underlies finite type invariants of virtual
knots/braids. (Compare with the well-understood relationship between
knots, chord diagrams and Lie algebras). Over the grant period I plan
to fully understand a universal theory of quantum groups as a natural
object within the context of virtual knot theory. See http://www.math.toronto.edu/~drorbn/Talks/Kyoto-0705/ and ---/Tianjin-0707/.
* Computations in Knot Theory and the Knot
Atlas. I Plan to continue contributing to the computational
package KnotTheory` and to the Knot Atlas. Both projects were founded
by me but by now have received contributions by many others (especially
S. Morrison). See http://katlas.org.

What about Khovanov homology? I made significant contributions
to the highly fashionable subject of Khovanov homology (in fact, while
Khovanov is definitely the father of the field, I share the credit for
making it fashionable...). Yet at the moment I don't feel mature enough
to study this topic any further. I'd rather "categorify" knot
invariants only after I properly understand the "algebra" on which they
ought to be defined (in the sense of my first project above). And how
can I
even start categorifying other aspects of the theory of quantum groups,
when in my opinion this theory in itself is so poorly understood (at
least in the sense of my second project)? With luck, at the end of this
grant period I will be ready to return to Khovanov homology and
categorification in general.

In science, though, the predicted is always less interesting than
the unpredictable. With luck, at least some of my work in the next five
years will be on topics I haven't yet heard of.